A large-scale particle system with independent jumps and distributed synchronization

Abstract

We study a system consisting of $n$ particles, moving forward in jumps on the real line. Each particle can make both independent jumps, whose sizes have some distribution, or ``synchronization'' jumps, which allow it to join a randomly chosen other particle if the latter happens to be ahead of it. System state is the empirical distribution of particle locations. The mean-field asymptotic regime, where $n\to\infty$, is considered. We prove that $v_n$, the steady-state speed of the particle system advance, converges, as $n\to\infty$, to a limit $v_{**}$ which can be easily found from a {\em minimum speed selection principle.} Also, as $n\to\infty$, we prove the convergence of the system dynamics to that of a deterministic mean-field limit (MFL). We show that the average speed of advance of any MFL is lower bounded by $v_{**}$, and the speed of a ``benchmark'' MFL, resulting from all particles initially co-located, is equal to $v_{**}$. In the special case of exponentially distributed independent jump sizes, we prove that a traveling wave MFL with speed $v$ exists if and only if $v\ge v_{**}$, with $v_{**}$ having simple explicit form; we also show the existence of traveling waves for the modified systems, with a left or right boundary moving at a constant speed $v$. Using these traveling wave existence results, we provide bounds on an MFL average speed of advance, depending on the right tail exponent of its initial state. We conjecture that these results for exponential jump sizes generalize to general jump sizes.